Hi Jamie,
First, there is nothing like a late-night pizza delivery after a long day exploring all that the theme parks have to offer! There have been many nights that my husband and I haven't even realized that we were hungry until that final ride "home" on the bus. Pizza always hits the spot!
Walt Disney World Resort pizza delivery operates at different times for the various Resorts, but you are correct that it begins at 4 P.M. Deluxe Resorts, Disney Vacation Club Resorts and Disney's Coronado Springs Resort do offer room service throughout the day if you are craving something other than pizza.
If you absolutely must have the perfect combination of dough, cheese, sauce and veggies that we all know and love, you can order a pizza through most of the national chains. Just keep in mind that you will need to meet the delivery person in the lobby to pick-up your order.
As a final note, please ignore any flyers that might be slipped underneath the door in your Disney's Old Key West Villa. Since solicitation is a direct violation of policy, these are rarely, if ever, legitimate. If you are ever doubt, give the front desk a call to confirm.
I hope that helps, Jamie!
Wishing you many magical memories,
Sally